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## G = M4(2).24C23order 128 = 27

### 6th non-split extension by M4(2) of C23 acting via C23/C22=C2

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — M4(2).24C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ 1+4 — M4(2).24C23
 Lower central C1 — C2 — C22 — M4(2).24C23
 Upper central C1 — C2 — C2×C4○D4 — M4(2).24C23
 Jennings C1 — C2 — C2 — C2×C4 — M4(2).24C23

Generators and relations for M4(2).24C23
G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2b, bab=a5, cac-1=a5b, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >

Subgroups: 724 in 378 conjugacy classes, 170 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4.D4, C4.10D4, C2×M4(2), C8○D4, C22×D4, C2×C4○D4, C2×C4○D4, 2+ 1+4, C2×C4.D4, M4(2).8C22, Q8○M4(2), C2×2+ 1+4, M4(2).24C23
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C23×C4, C22×D4, C22×C22⋊C4, M4(2).24C23

Permutation representations of M4(2).24C23
On 16 points - transitive group 16T200
Generators in S16
```(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 5)(3 7)(9 13)(11 15)
(1 16 7 10 5 12 3 14)(2 13 4 11 6 9 8 15)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)```

`G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(9,13)(11,15), (1,16,7,10,5,12,3,14)(2,13,4,11,6,9,8,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)>;`

`G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(9,13)(11,15), (1,16,7,10,5,12,3,14)(2,13,4,11,6,9,8,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16) );`

`G=PermutationGroup([[(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(3,7),(9,13),(11,15)], [(1,16,7,10,5,12,3,14),(2,13,4,11,6,9,8,15)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16)]])`

`G:=TransitiveGroup(16,200);`

41 conjugacy classes

 class 1 2A 2B ··· 2H 2I ··· 2N 4A ··· 4H 4I 4J 8A ··· 8P order 1 2 2 ··· 2 2 ··· 2 4 ··· 4 4 4 8 ··· 8 size 1 1 2 ··· 2 4 ··· 4 2 ··· 2 4 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 1 2 8 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 D4 M4(2).24C23 kernel M4(2).24C23 C2×C4.D4 M4(2).8C22 Q8○M4(2) C2×2+ 1+4 C22×D4 C2×C4○D4 C4○D4 C1 # reps 1 6 6 2 1 12 4 8 1

Matrix representation of M4(2).24C23 in GL8(ℤ)

 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0
,
 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0
,
 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0

`G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;`

M4(2).24C23 in GAP, Magma, Sage, TeX

`M_4(2)._{24}C_2^3`
`% in TeX`

`G:=Group("M4(2).24C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1620);`
`// by ID`

`G=gap.SmallGroup(128,1620);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,2028,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^8=b^2=d^2=e^2=1,c^2=a^2*b,b*a*b=a^5,c*a*c^-1=a^5*b,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^4*d>;`
`// generators/relations`

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